Ana Moura sits in a restaurant in downtown Lisbon explaining how she was always destined to be a fado singer, she just hadn't always realised it. "We have this saying in Portugal: you have to be born a fado singer, it's not something you just learn."

Moura is today one of the best-known fado singers in the world, following in the footsteps of Mariza and her global success. She has sung with both the Rolling Stones and Prince, who is a major fan. And yet it was almost by accident that she found herself singing fado – the plaintive and melancholic style of song that is often known as the Portuguese blues.

"Things in my life just happen," she says.

Brought up in a musical household, her parents "realised that I had a nice voice" when she was just four, and she was six when she first sang fado in public, at a wedding. Later, as a teenager in Lisbon, her singing teacher told her a well-known producer was looking for a pop-rock singer, and asked if she would audition. She was chosen, and found herself recording an album with "a rock band with violins, like Portishead but more rough". In the studio, the producer one day asked her if she sang fado. She obliged, and found that "my posture was suddenly different, and so was the way I sang". "That," she was told, "is how you should sing rock – with intensity." The rock album was never released "because there was a big mess with the record company".

Visiting a bar soon after with friends, she found that they had arrived "by coincidence" on a fado night, "though this was not a fado house". It was suggested she should sing, and she so impressed the fadistas present that other invitations followed, leading her to Senhor Vinho, a typical fado house "with a mysterious environment, a small room with candles, no microphones and great food. The diners are completely quiet when the singers come on." Here she learned improvisation and the fado singer's skill of finding new poems to set to classic traditional fado melodies. Here she also met the guitarist, composer and lyricist Jorge Fernando, who had worked with the greatest fadista, the legendary Amália Rodrigues, and who would later become Moura's producer. "He told me I had to do fado, I was born a fadista. It was a great compliment from the roots."

What those musicians all recognised was Moura's elusive quality, so crucial to fado and other emotional musical styles – soul. Playing at Womad this summer, backed by a traditional fado trio of Portuguese guitar, classical guitar and bass, she won over the crowd with her intense, emotional but never over-dramatic treatment of songs that ranged from traditional fado to an exquisite and unexpected fado treatment of the Rolling Stones's No Expectations.

She is shy off-stage, "like many artists I have met who play for many people", but she has always been determined to do things her way. There was a time, she says, when she was "surrounded by a lot of people who had their ideas of what an artist should be, and I don't think there are rules for that. It should come from inside. They wanted me to have a different personality on stage, and told me what I should say, but I hate things that are programmed. We are authentic if we sing with our souls. I want the spontaneity that my parents taught me when I was young."

The musical spontaneity also helped her experiment with so-called musical fado, in which other types of songs are given a fado treatment "and what gives it the fado smell is the interpretation – a great fado singer can make even Mozart sound like fado". So her albums have mixed traditional and musical fado; on her last studio set Leva-Me Aos Fados she recorded one track with a band, Gaiteiros De Lisboa, "who create their own instruments and play rhythms that remind me of the folk music of the north of Portugal". More recently, she has performed fado songs with the Frankfurt Radio Big Band, "a completely different approach with a jazz flavour."

Her experimental attitude has also helped her establish links with rock superstars. Rolling Stones' saxophonist Tim Ries invited her to sing on Rolling Stones Project 2, on which international artists (including Milton Nascimento and Bill Frisell) interpret Stones songs. Moura recorded No Expectations – "a great song with a great melody from Keith. It works really well as a fado", and also Brown Sugar, done as a rhythmical fado mouraria – "a roots fado", as she describes it.

When the Stones came to Lisbon in 2007 she invited them to a fado house "where Mick Jagger was filming everything". The next day she was asked to their concert at the José Alvalade football stadium. On arrival, she found herself invited to sing live with them. "There was just a quick rehearsal, with Jagger singing in a higher key than I expected. I had to improvise".

Then there's Prince, "who bought my albums on the internet and came to see me sing in Paris – he arrived in a red suit with a sparkling cane. He's very nice." He too invited her on stage in Lisbon – but Prince played fado, with an electric-guitar intro to A Sós Com A Noite, "and it was magical. All the Portuguese went crazy, then when he started to play everybody went quiet – people had the sensation that they went to another level."

Since then, she and Prince have played together at his studio in Minneapolis and at her house outside Lisbon. "Sometimes our tours coincide and we do some jams." So will any of those recordings appear on her next album? She's not saying. "You never know … let's see what comes."